Porto Ercole, M&MKT 2016 Multiscale Systems from Particles to Continuum: Modelling and Computation Modelling Muscle Contraction a multiscale approach Giovanni Naldi Dipartimento di Matematica ``F. Enriques Università di Milano
Skeletal Muscle Striated and voluntary Attaches to skeleton via tendons Most abundant tissue in the body (45-75% of body weight)
Structure of a muscle cell A. Fascicles fiber bundles B. Fibers muscle cell bundles of myofibrils C. Myofibrils D. Sarcomeres (series) E. Actin & Myosin Filaments
Fascicles A muscle is composed of multiple fascicles in parallel A sheath of connective tissue surrounds the muscle (epimysium) Each fascicle is surrounded by connective tissue (perimysium) Fascicles composed of bundles of muscle fibers
Muscle Fiber Long, cylindrical, multinucleated cells Between fibers are blood vessels Surrounded by endomysium Composed of myofibrils
Literally (muscle thread) Contractile element of muscle Made up of filaments aligned in parallel filaments make striations - Banding pattern One repeating unit is called a sarcomere string of sarcomeres in series sarcomere Myofibrils A-band I-band filament Z-line
Sarcomeres Functional unit of muscle contraction Literally muscle segment Number of sarcomeres in a fiber is very important to muscle function When each sarcomere shortens the same amount, the fiber with more sarcomeres will shorten more. Made up of myofilaments Thick and thin filaments A sarcomere is about 1.5-3.5 m long.
Myosin Composed of two heavy chains twisted together. Has a head and a tail. The myosin head has 2 binding sites for actin and ATP. The head can swivel about its hinge. They bind to actin filaments to form cross- bridges to span the space between filaments. In skeletal muscle, about 250 myosin molecules create to make a thick filament.
Actin Multiple actin molecules (G-actin) polymerize to form a long chain (F-actin). A pair of F-actin chains twist together to form the thin filament. G-actin has a binding site for a myosin head.
Sliding filament theory Muscle contraction AF Huxley and HE Huxley (1954, Nature) Light and Electron microscopy Both published results same time in Nature Does not explain lengthening contractions The exertion of force by muscle is accompanied by the sliding of thick and thin filaments past one another Commonly explained by cross-bridges (AF Huxley, 1957) Video
Sarcomere Length Changes During Contraction Sir Andrew Huxley The I-band and H-zone shorten but the A-band remains constant. Sliding-filament theory of contraction. Overlapping filaments slide past each other.
Muscle Force Depends (mainly) on Four Factors Sarcomere (muscle) length Velocity of muscle contraction Activation level Previous contraction history Force-Length Relationship Fact: Muscles at very long and very short lengths can not produce high forces Fact: Maximal force production of a muscle depends on its length
Force-Length Relationship
Sliding filament model, 1954, hypothesis No deformation of filaments during the contraction Only the distance between the Z disc changes
Sliding filament model (cross bridges) According to the sliding filament theory, the contractile component is able to develop force or to shorten because of the interaction between the proteins belonging to the two sets of filaments, actin and myosin (crossbridge cycle) The cross-bridge cycle is dependent on the level of activation of the contractile proteins. The activation is considered to be proportional to the amount of Ca ++ bound to troponin belonging to the thin filament. The amount of bound Ca ++ can be obtained as output of a compartmental system that describes the Ca ++ movements inside the muscle cell. The input of the model is the action potential, which affects some parameters of the Ca ++ compartmental model. Here, for simplicity, we consider a two state model for the cross-bridges
Rheological model (Eisenberg- Hill) According to the classic view of Hill the muscle s mechanical properties can be separated into three elements. Two elements are arranged in series: (CE) an active force generating contractile freely extensible at rest, but capable of shortening when activated; (SE) a series elastic (i.e. depending on the amount of strain) or viscoelastic (i.e. depending also on strain rate), which represents the structures on which the CE exerts its force during contraction (tendons, Z-band, connective tissue). To account for the mechanical behavior of muscle at rest, a parallel (visco-)elastic element (PE) is added (sarcolemma, collagen or elastic fibers).
Contractile Element: Half -Sarcomere Let u(x,t) denote the density of cross-bridges which. at time t, are halfattached between myosin and actin at the distance x from the equilibrium position in the element CE. x actin The dynamics of the cross-bridges population is given by myosin Cross bridge du dt (x,t) = x, t, u where gives the balance of the formation and the breakdown of bridges. By considering a fixed frame, the material derivative becomes du dt u (x,t) = t +dx dt u x = x, t, u where v(t)= dx/dt is the speed of the contraction of the half sarcomere.
Assuming, for simplicity, a linear elastic behavior with stiffness k, we have that the force developed at time t by all interacting cross-bridges in the half-sarcomere is given by: The PE and SE elastic forces are F PE t = G 1 L PE t L PE0 F SE t = G 2 L SE t L SE0 where G 1 and G 2 are given nonlinear one-to-one real maps, and L PE0, L SE0 are the initial lengths (L PE0 (t)=l(t))
An example for G 1 and G 2 has been experimentally obtained by Capelo, Comincioli, Minelli, Poggesi, Reggiani, and Ricciardi (J. Biomech. 1981) Let us remark that we are also now assuming that the actin and myosin filaments do not deform. For the forces, we have P(t) = F PE (t) + F SE (t), F CE (t)=f SE (t) where P(t) denotes the muscle tension.
We can consider different experimental situations: isometric, L(t) as input and P(t) as output; isotonic, P(t) as input and L(t) as output; and isometric-isotonic, sequence of the previous situations. Denoting by v(t) the rate of contraction we have (by using the parameter identification of CCMPRR) Limiting ourselves to the isometric case with L(t) = const., we obtain,
Note. Here we write v(t)=z (t)
Theorem. The ISOMETRIC PROBLEM has one and only one solution. Outline of the proof. Step 1 Consider the hyperbolic problems with z(t) given (for the moment): the characteristics technique allows us to find a unique solution Uz (x, t) (depending obviously on z ) such that Step 2 By means of the data assumptions one can state the a priori boundedness of the solutions z(t), and Uz
Step 3 Given function z(t), z(0), we can compute Uz(x,t), then (by using the definition of z(t) and integrating the hyperbolic equation), Step 4 Solve the problem: find z(t) in C 0,1 ([0,T]) such that z(t) = Gz(t) t [0,T] By using a suitable fixed point Theorem.
Note. The cross bridges could be in different states (and produce different forces), we have a system of uj(x,t) with j=1,,m with a transition matrix.
The whole picture: From the input Stimulus to the Mechanobiological transduction
Heterogeneity has been shown in skeletal muscle, but, how the heterogeneity of functional properties inside each fibre determines the behaviour of the fibre taken as a whole, and how the heterogeneity among adjacent fibres in a muscle can influence the contractile performance of that muscle taken as a whole. Series Parallel
Example, from P. Colli, V. Comincioli, G.N., C. Reggiani, 1988.
Open problems: - Analysis of the networks of the sarcomere - Efficient numerical methods - Parameter identification and experiments -.